Quadratics: How a Function-Based
Approach Can HelpQuickTime™ and a
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http://generative.edb.utexas.edu/materials/aisd09/aisd10.htm
Today
• FBA can help with distinguishing when/how to use rules for simplifying from rules for solving in a way that students can check themselves.
• Go over FBA activities that can be used “tomorrow” in class to introduce quadratics
• Make connection from solving with quadratics back to solving with linear functions.
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Aug
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But first some data about the kinds of algebra-related challenges the function-based approach can address
The survey was given to 80 Algebra 1 students at a local high school on November 24, 2009. All students were taught by the same teacher. 63 of the 80 students completed the survey.
Research from Prudie Cain
Question #1: Simplify 3x+2-2x-1
0%10%20%30%40%50%60%70%80%90%
100%
combine like terms apply the distributiveproperty
do the same thing toboth sides
cancel multiply both terms by it do the opposite substitute a number forx
Examine graphs using acalculator or computer
Examine tables using acalculator or computer
Examine graphs usingpencil/pen or paper
Examine graphs usingpencil/pen or paper
Definitely would use this strategyLikely to use this strategyUnlikely to use this strategyDefinitely would NOT use the strategyNo Response
Question #2: Simplify -7(x+2)+4
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
combine like terms apply thedistributive
property
do the same thingto both sides
cancel multiply both termsby it
do the opposite substitute anumber for x
Examine graphsusing a calculator
or computer
Examine tablesusing a calculator
or computer
Examine graphsusing pencil/pen or
paper
Examine graphsusing pencil/pen or
paper
Definitely would use this strategy
Likely to use this strategy
Unlikely to use this strategy
Definitely would NOT use the strategy
No Response
Question #3: Solve 2x-5=x+4
0%10%20%30%40%50%60%70%80%90%
100%
combine like terms apply the distributiveproperty
do the same thing toboth sides
cancel multiply both termsby it
undo the operation(for example: undo
the division)
substitute a numberfor x
Examine graphsusing a calculator or
computer
Examine tablesusing a calculator or
computer
Examine graphsusing pencil/pen or
paper
Examine graphsusing pencil/pen or
paper
Definitely would use this strategyLikely to use this strategyUnlikely to use this strategyDefinitely would NOT use the strategyNo Response
Question #4: Solve x-4-3x=3+2x+1
0%10%20%30%40%50%60%70%80%90%
100%
combine like terms apply the distributiveproperty
do the same thing toboth sides
cancel multiply both termsby it
do the opposite substitute a numberfor x
Examine graphsusing a calculator or
computer
Examine tablesusing a calculator or
computer
Examine graphsusing pencil/pen or
paper
Examine graphsusing pencil/pen or
paper
Definitely would use this strategy
Likely to use this strategy
Unlikely to use this strategy
Definitely wouldNOT use the strategyNo Response
Question #1: Simplify 3x+2-2x-1
0%10%20%30%40%50%60%70%80%90%
100%
combine like terms apply the distributiveproperty
do the same thing toboth sides
cancel multiply both terms by it do the opposite substitute a number forx
Examine graphs using acalculator or computer
Examine tables using acalculator or computer
Examine graphs usingpencil/pen or paper
Examine graphs usingpencil/pen or paper
Definitely would use this strategyLikely to use this strategyUnlikely to use this strategyDefinitely would NOT use the strategyNo Response
Goodnews
SizableFraction (~1/3) Select “Solving” Strategies
Most seem unaware of how calculator (FBA) might help
More likely to view paper version as useful
See thesestrategies asmeaningful
Question #2: Simplify -7(x+2)+4
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
combine like terms apply thedistributive
property
do the same thingto both sides
cancel multiply both termsby it
do the opposite substitute anumber for x
Examine graphsusing a calculator
or computer
Examine tablesusing a calculator
or computer
Examine graphsusing pencil/pen or
paper
Examine graphsusing pencil/pen or
paper
Definitely would use this strategy
Likely to use this strategy
Unlikely to use this strategy
Definitely would NOT use the strategy
No Response
Goodnews
Still sizablefraction (~1/3) Select “Solving” Strategies
And FBA is seen as even less useful
Although seemingly more useful or acceptable if done with paper and pencil
Question #3: Solve 2x-5=x+4
0%10%20%30%40%50%60%70%80%90%
100%
combine like terms apply the distributiveproperty
do the same thing toboth sides
cancel multiply both termsby it
undo the operation(for example: undo
the division)
substitute a numberfor x
Examine graphsusing a calculator or
computer
Examine tablesusing a calculator or
computer
Examine graphsusing pencil/pen or
paper
Examine graphsusing pencil/pen or
paper
Definitely would use this strategyLikely to use this strategyUnlikely to use this strategyDefinitely would NOT use the strategyNo Response
Goodnews
And FBA is EVEN LESS USEFUL for solving than for simplifying
Still sizablefraction (~1/3) seems to select an inappropriate or
irrelevant strategy
Yet don’t seem to be as able to connect to potentially relevant sub-strategies
Although seemingly more appropriate if done with paper and pencil
Question #4: Solve x-4-3x=3+2x+1
0%10%20%30%40%50%60%70%80%90%
100%
combine like terms apply the distributiveproperty
do the same thing toboth sides
cancel multiply both termsby it
do the opposite substitute a numberfor x
Examine graphsusing a calculator or
computer
Examine tablesusing a calculator or
computer
Examine graphsusing pencil/pen or
paper
Examine graphsusing pencil/pen or
paper
Definitely would use this strategy
Likely to use this strategy
Unlikely to use this strategy
Definitely wouldNOT use the strategyNo Response
Goodnews
Still sizablefraction (~1/3) seems to select an inappropriate or
irrelevant strategy
But somehow it is more likely to be useful if you graph or create tables using pencil and paper
And FBA not seen as useful
Again, seemingly not connected to potentially relevant sub-strategies
Simplifying and Solving are Confused
Function-Based Approach
(1) Can Help (see images)
(2) Can Help in Ways that Will Help with Future Challenges
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Today
• FBA can help with distinguishing when/how to use rules for simplifying from rules for solving in a way that students can check themselves.
• Go over FBA activities that can be used “tomorrow” in class to introduce quadratics
• Make connection from solving with quadratics back to solving with linear functions.
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Big Picture(FBA)
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For lines, solving for xmeans getting to a machine that gives out what you put in, knowing what you get out and thereby knowing what you put in.
x
79.8568
x
?
3?
IN
OUT
(Must be 3)
79.8568
e.g., X = 3
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unsolving_lines_00.ggb[1]
unsolving_lines_01.ggb[2]
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2nd TRACECALC
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GRAPH
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unsolving_lines_02.ggb
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ENTER ENTER
ENTER
2nd MODEQUIT
[1]
[2]
[3]
2nd PRGMDRAW
ENTER
ENTER
GeoGebra
Type3orx
Calculator
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Click to turn on
Enter each pair (side) in solving
Example of “unsolving”; Can do (almost) anything to both sides[e.g., multiply both sides by x]
Repeat [1] - [3]
or
(because after using 5:intersect the x is set to the x-value of the intersection)
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Can turn off functions by driving over equal sign and pressing ENTER
If intersection happens off the screen use WINDOW
unsolving_lines_03.ggb
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Each transformation changes lines but preserves solution
Unsolving … Do whatever you want to both sides.
unsolving_lines_03.ggb
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• Check intersection (same x-value?)• Draw vertical line at this value• Put new pair of functions on the board
For quadratics, solving for xmeans getting to a place where what we know about roots helps us know the original intersection(s).
Thankfully, doing the same thing to both sides does the same sort of thing to the original functions (replaces the original pair with functions that are NOT equivalent but do have the same solution set).
(Use other activities to help understand the importance of roots)
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[1]
[2]
[3]
solving_quads_00.ggb
solving_quads_01.ggb
Calculator:
Follow earlier steps and use 5:intersect and draw 4:Vertical
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solving_quads_02.ggb
Draw vertical lines (& adjust appearances)
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What Do We Notice?
Like for lines, can we do (almost) ANYTHING to both sides? [Unsolving] Try it. What happens?
solving_quads_03.ggb
Why is this good?: We know how to find roots.
solving_quads_03plus.ggb
Just like for lines, enter each pair (side) in solving
Use 5:Intersect to find x-values for drawing vertical lines
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Where could we go from here?Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values.
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solving_quads_04.ggb
Challenge
handouts
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unsolving_lines_00.ggb[1]
unsolving_lines_01.ggb[2]
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2nd TRACECALC
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GRAPH
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unsolving_lines_02.ggb
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ENTER ENTER
ENTER
2nd MODEQUIT
[1]
[2]
[3]
2nd PRGMDRAW
ENTER
ENTER
GeoGebra
Type3orx
Calculator
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Double-click on object to change color or thickness (style)
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Click to turn on
Enter each pair (side) in solving
Example of “unsolving”; Can do (almost) anything to both sides[e.g., multiply both sides by x]
Repeat [1] - [3]
or
(because after using 5:intersect the x is set to the x-value of the intersection)
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QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
Can turn off functions by driving over equal sign and pressing ENTER
If intersection happens off the screen use WINDOW
unsolving_lines_03.ggb
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Each transformation changes lines but preserves solution
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[1]
[2]
[3]
solving_quads_00.ggb
solving_quads_01.ggb
Calculator:
Follow earlier steps and use 5:intersect and draw 4:Vertical
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What Do We Notice?
Like for lines, can we do (almost) ANYTHING to both sides? [Unsolving] Try it. What happens?
solving_quads_03.ggb
Why is this good?: We know how to find roots.
solving_quads_03plus.ggb
Just like for lines, enter each pair (side) in solving
Use 5:Intersect to find x-values for drawing vertical lines
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Where could we go from here?Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values.
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solving_quads_04.ggb
Challenge
Where could we go from here?Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values.
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solving_quads_04.ggb
Challenge
Where could we go from here?Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values.
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solving_quads_04.ggb
Challenge
Where could we go from here?Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values.
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solving_quads_04.ggb
Challenge
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